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Is it circular to define the Von Neumann universe using “sets”?

I was just reading the Wikipedia page on the Von Neumann universe, where it is stated that this universe "is often used to provide an interpretation or motivation of the axioms of ZFC." However, later ...
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Meaning of the word “axiom”

One usually describes an axiom to be a proposition regarded as self-evidently true without proof. Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises ...
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Cauchy's real line and math philosophy till XIX

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ ...
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Is Leibnizian calculus embeddable in first order logic?

We just published an article making what we feel is a plausible case in favor of an affirmative answer in Foundations of Science, see preprint here. The basic argument is that while such a requirement ...
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Have humans ever used the Log Scale convention in the past rather than the Linear one?

There are many examples where our senses are based off of log scales such as volume of a noise, ability to guess (i.e.) plus or minus a power of 10 with Fermi, and even when we measure pain on 1 to 10 ...
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On “why” questions in mathematics

In response to the question How would one be able to prove mathematically that $1+1 = 2$?, Asaf Karagila explains: In a more general setting, one needs to remember that $0,1,2,3,…$ are just ...
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On logic vs information theory

If the statements All crows are black and All non black things are non crows are equal, then why is the former so much easier to communicate by giving examples? What implications does this ...
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Is formal truth in mathematical logic a generalization of everyday, intuitive truth?

I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth. Personal background: When I ...
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What questions or areas in the foundations of mathematics remain active research fields?

By foundations of mathematics I am referring to the mathematical, logical, and philosophical foundations of the subject. I'm interested in seeing which of these have active research going on within ...
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Are all statements about math inherently formal? Can one do math without formal logic? [duplicate]

Are all people who do mathematics applying (whether they know it or not) formal logic? Does every statement someone may make about math, at its core, a formal statement in mathematical logic? ...
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Different Mathematics

Hey I am a high school student who is very interested in the philosophy of mathematics. I was watching this talk by Stephen Wolfram about whether or not mathematics is invented or discovered. In it he ...
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A question about the real line and the Dirichlet function.

Though the graph of the Dirichlet function is non-drawable, I think if we have to draw it in some informal way then it will be two complete lines (instead of isolated points). Here's my reasoning: ...
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Should we or should we not take $1$ as a prime number? [duplicate]

I think I know that there were times in the past when it was convenient to look at a number $1$ as a prime number, and, as far as I can remember, even then it was dependent on who we ask is it prime ...
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Division of segments into infinitely many parts.

Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2. If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested ...
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mathematical terms with fractions and variables - usage in daily life?

what usage do algebraic fractions (monomial or polynomial) have in our life? Are there specific professions that deal with them now and then? Where exactly in technology do they have their very own ...
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$Con(T) + T\vdash \neg\neg A$ implies $Con(T+A)$ for any intuitionistic theory T

It's easy to notice that for any intuitionistic theory T: $Con(T) + T\vdash \neg\neg A$ implies $Con(T+A)$ $Con(T) + T\vdash \neg\forall x\neg A$ implies $Con(T+\exists x A)$ where $Con(T)$ means, ...
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Scalar multiplication as a special form of matrix multiplication

Question What do we gain or lose, conceptually, if we consider scalar multiplication as a special form of matrix multiplication? Background The question bothers me since I have been reading about ...
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Impossibility of proving a foundation to be consistent

An argument came to my mind that seems really mind-blowing and I haven't found it anywhere. Here's how it goes: We call a formal system F embodied in classical logic a foundation of mathematics when ...
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Quasi mathematical objects [closed]

I was looking on this post http://www.songho.ca/math/euler/euler.html and I came to the comment that says "i is not a number at all. It is an ill-formed concept. There is a vast difference between a ...
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Could relational operators be used to construct formal theory of natural numbers which is “stronger” than Peano Axioms?

This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of ...
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Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
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Developments from Charles Peirce's logic diagrams?

These last weeks I have been revisiting Charles Sanders Peirce's logical or thought diagrams (what he called, alpha, beta and gamma diagramms) and I found many of them highly interesting. Some ...
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What exactly are the numbers we use everyday?

Pi can be defined as diameter / circunference of a circle. But what is a circle? You can't tell a computer: "build a circle and divide its diameter by its ...
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What is the simplest mathematical object? [closed]

What is the simplest mathematical object? I am talking about mathematics in the most abstract way possible, and not as some concrete axiomatic theory (e.g. foundational ones, like ZFC). After a lot ...
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How can I learn Math intuitively? [closed]

I am currently a Junior in High School. I am in an Intermediate Algebra class, but my teacher does not always explain things in a way I can understand. I like to learn Math intuitively, but my teacher ...
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How should a mathematically-inclined person learn descriptive statistics?

I am interested in learning descriptive statistics. But I am completely baffled, that there seem to be no mathematically rigorous books on this subject, as far as I know at least. The Wikipedia page ...
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Why might Dieudonne have been “begging the question” by appealing to second-order Peano Axioms?

Following a comment by Peter Smith, I've been reading A. R. D. Mathias's paper The Ignorance of Bourbaki. Parts of the paper are above my head, but I understand it well enough for my own amateurish ...
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Why can't a Hilbert curve be used to put the real numbers into a listable format?

There's a very good chance this question will make absolutely no sense, as my understanding of Hilbert curves is very superficial. But let me explain where my question is coming from. From my ...
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Can the tehniques of higher level mathematics solve most of Olympiad level math problems through straighforward applications?

Working through many Olympiad math problems(pre-undergrad) I've found that simple applications of undergrad math will solve many of them. Does this trend go on? Can it be that Putnam problems are ...
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No Proof, Just Luck

I just read about the Goldbach Conjecture and it got me thinking about probabilities. Supposing that prime numbers are somewhat randomly distributed) then if we calculate the odds of a given even ...
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Natural numbers, divisors, primes and their generalized means

Let div, nat and pri the finite sequences given in increasing order for an integer $n\geq 1$ of its divisors $1=d_1<d_2<\ldots d_{\sigma_0(n)}=n$, the first $n$ natural numbers, and the first ...
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How can we recognize if something is a number?

There are formal definitions of various types of numbers; natural numbers, real numbers, ordinal numbers, cardinals etc. And we all regard them as some type of number. Are there properties that are ...
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What is this ontological position called?

If one believes that certain 'abstract' mathematics-like concepts do exist, yet the mathematics we construct and develop as humans are only approximations of those real concepts, approximations shaped ...
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How does one refute this ultrafinitist argument?

From Wikipedia: Edward Nelson criticized the classical conception of natural numbers because of the circularity of its definition. In classical mathematics the natural numbers are defined as $0$ ...
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Can I just make this function up?

The Lambert W function was made to solve the problem $xe^x=k$ for $x$, which is given as $x=W(k)$. Could I just make a function $x=F(k)$ which solves $x\cos(x)=k$? Even though the solution has an ...
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Intuitonism and metamathematics.

There are various reasons why one would want to reject the law of the excluded middle when doing "normal" mathematics, which I won't get to here, but accepting those, does the same reasoning hold when ...
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Group Theory and Cardinals [duplicate]

I was wondering the following: Let's suppose that $G$ is a non-empty set, then, can we always find a binary operation $*$ such that $(G,*)$ is a group? For example, if we fix $G=\mathbb{Q}$ the sum ...
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Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not ...
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Creating mathematics vs building houses

I found the following quote in the book "Calculus" by Michael Spivak. (At the first page of Part 5,Epilogue, where he will discuss fields, construction of the real number, and uniqueness of the real ...
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Implication as defined in mathematical logic [duplicate]

If we can take the truth of an implication to mean the validity of reasoning, then that would mean that all reasoning that begins with false premises is valid reasoning. Are there no counterexamples ...
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An intuitive explanation for the negatives of divergent summations?

I am looking for an intuitive explanation for why divergent summations (that are always increasing) have finite values assigned as negative. An example that is beyond "because the math says so" kind ...
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Prime Formulas in Heyting Arithmetic

I have been reading into Intuitionistic Logic, namely Heyting arithmetic, and I've bumped into this: Corollary 3.9. Let $A_0$ be a quantifier-free formula of $\mathscr L(HA)$. Then $$HA\vdash ...
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Should it be allowed to apply classical logic to set theory?

It is well known, that the generalized continuum hypothesis isn't provable from the standard axiom system ZFC. GCH (generalized continuum hypothesis). For every infinite set A, there isn't a set M ...
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Evolution of Definitions

I try to understand how the definitions of mathematics have evolved (or formulated)... I'll use the epsilon-delta continuity definition as an example to ask my question... It may seem trivial, but ...
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What does it mean to solve an equation?

This question might be more philosophical than mathematical. In school we are taught how to solve equations such as $x^2 - 1 = 0$ or $\sin(x) - 1= 0$. Solutions to these equations are quite simple. ...
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How much of (pure) mathematics is first order logic?

Most automated theorem provers are built for first order logic only. How much are they missing out by focusing on first order logics?
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Did Ackermann produce a finitary consistency proof of second-order $PRA$?

In Wilhelm Ackermann's Doctoral Thesis (it is claimed, by Richard Zach, for one, in his paper "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program", arXiv: ...
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When can independence of a statement in a theory be reduced to “truth”?

Since the Goldbach conjecture is in $\Pi_1^0$, if it were proven to be independent of Peano Arithmetic, it would follow that the Goldbach conjecture is true (i.e. true in the standard model), since ...
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Where do models of set theory live?

When we are studying independence proofs we are dealing with statements of the form $Cons(T)\rightarrow Cons(T')$ where $T$ and $T'$ are first order theories; commonly $T$ and $T'$ are subtheories of ...
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Does it make any sense to prove $0.999\ldots=1$?

I have read this post which contains many proofs of $0.999\ldots=1$. My question is, Does it make any sense to prove this equality? Can one give any "meaning" of the symbol $0.999\ldots$ ...